MinT - Best Known (72, 72+175, s)-Nets in Base 4

Best Known (72, 72+175, s)-Nets in Base 4

(72, 72+175, 66)-Net over F₄ — Constructive and digital

Digital (72, 247, 66)-net over F₄, using

t-expansion [i] based on digital (49, 247, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(72, 72+175, 105)-Net over F₄ — Digital

Digital (72, 247, 105)-net over F₄, using

t-expansion [i] based on digital (70, 247, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(72, 72+175, 488)-Net in Base 4 — Upper bound on s

There is no (72, 247, 489)-net in base 4, because

1 times m-reduction [i] would yield (72, 246, 489)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13215 709533 025642 900788 555029 449356 110935 937216 797879 780876 341440 154456 815602 917368 020235 728831 239142 183395 481595 978603 882799 794128 982569 420247 615920 > 4²⁴⁶ [i]